List of top Mathematics Questions on Binomial theorem asked in KEAM

The middle term in the expansion of \( \left(\frac{10}{x} + \frac{x}{10}\right)^{10 \) is:}

The coefficient of \( x^{49} \) in the product \( (x-1)(x-2)(x-3) \dots (x-50) \) is:

The sum of the coefficients in the binomial expansion of \( \left( \frac{1}{x} + 2x \right)^6 \) is equal to:

If $ n=5, $ then $ {{{{(}^{n}}{{C}_{0}})}^{2}}+{{{{(}^{n}}{{C}_{1}})}^{2}}+{{{{(}^{n}}{{C}_{2}})}^{2}}+..... $ $ +{{{{(}^{n}}{{C}_{5}})}^{2}} $ is equal to

In the expansion of $ {{(1+x+{{x}^{2}}+{{x}^{3}})}^{6}}, $ the coefficient of $ {{x}^{14}} $ is

The coefficient of $x$ in the expansion of $ (14+x)(1+2x)(1+3x)....(1+100x) $ is

The coefficient of $x ^{49}$ in the product $\left(x-1\right)\left(x-2\right)\cdots\left(x-50\right)$ is

The term independent of $x$ in the expansion of $\left(x+\frac{1}{x^{2}}\right)^{6}$ is

The remainder when $2^{2016}$ is divided by $63$, is

Let $t_n$ denote the $n^{th}$ term in a binomial expansion. If $ \frac{t_{6}}{t_{5}}$ in the expansion of $(a+ b)^{n+4}$ and $ \frac{t_{5}}{t_{4}}$ in the expansion of $(a + b)^n$ are equal, then $n$ is

$ ^{15}{{C}_{0}}{{.}^{5}}{{C}_{5}}{{+}^{15}}{{C}_{1}}{{.}^{5}}{{C}_{4}}{{+}^{15}}{{C}_{2}}{{.}^{5}}{{C}_{3}}{{+}^{15}}{{C}_{3}}{{.}^{5}}{{C}_{2}} $ $ {{+}^{15}}{{C}_{4}}{{.}^{5}}{{C}_{1}} $ is equal to

The coefficient of $x^5$ in the expansion of $(1 + x^2)^5(1 + x)^4$ is

The coefficient of $ {{a}^{5}}{{b}^{6}}{{c}^{7}} $ in the expansion of $ {{(bc+ca+ab)}^{9}} $ is

If $ |x|<1, $ then the coefficient of $ {{x}^{6}} $ in the expansion of $ {{(1+x+{{x}^{2}})}^{-3}} $ is

If $C_{0}$, $C_{1}$, $C_{2}$, $C_{3}$, $\cdots$ are binomial coefficients in the expansion of $(1 + x)^n$, then $\frac{C_{0}}{3}- \frac{C_{1}}{4}+ \frac{C_{2}}{5}- \frac{C_{3}}{6}+ \cdots$ is equal to

If $\left(1+ax\right)^{n} =1+6x+\frac{27}{2}x^{2}+\cdots+a^{n}\, x^{n}$, then the values of $a$ and $n$ are respectively

If $\left(1+x+x^{2}\right)^{n} =1+a_{1}x+a_{2}x^{2} +\cdots+a_{2n}x^{2n}$, $2a_{1} -3a_{2} +\cdots-\left(2n+1\right)a_{2n}$ is equal to